Question

How do you solve for ${{\log }_{4}}x=3$ ?

Answer 1

In order to find the solution to this problem, we have to change the logarithm into an exponential and by taking the base 4 from the left and by pushing up 3 on the right to the power of 4. We can do this by converting in terms of exponential function.

Complete step by step solution:
According to our question, we have equation as:
$\Rightarrow {{\log }_{4}}x=3$
As we know logarithmic functions are the inverse of the exponential functions with the same bases, so we will apply the following formula,
${{\log }_{b}}x=y$ if and only if ${{b}^{y}}=x$
With this, we will evaluate our problem in terms of formula, so we will convert our terms in exponential function.
Therefore, we get:
$\Rightarrow {{\log }_{4}}x=3$
And we can write it as
$\Rightarrow x={{4}^{3}}$
Therefore, by solving cube we get:
$\Rightarrow x=64$
With this we get our solution of our problem:
$\Rightarrow {{\log }_{4}}64=3$
Therefore, from the above expression, we get the value of x as 64 which is the final solution.

Note: There is a common mistake which we tend to make with logarithms because when working with exponents, we have to work exponents in reverse and this is challenging for us, since we are not often so confident with the powers of numbers and the exponent properties.
Therefore, we have to be sure about the powers, just count the number of zeros to the right of the $1$ for positive exponents and move the decimal to the left for negative exponents.